Question

Arnold has a $50 gift card to use at a restaurant. She does not have any additional money to

spend at the restaurant. Arnold will purchase a drink that costs $5 and x number of food items
that cost $10 each. The function f(x) = 45 - 10x models the balance on the gift card after Arnold
makes the purchases. Which is the most appropriate domain of the function?

Answer 1

The domain of a function represents the set of all possible values for the independent variable (in this case, "x") that make sense within the context of the problem.

In this scenario, Arnold has a $50 gift card and wants to buy a drink for $5 and "x" food items for $10 each. The function f(x) = 45 - 10x models the balance on the gift card after these purchases. To find the appropriate domain, we need to consider two factors:

1. Arnold can't spend more money than he has on the gift card. Therefore, the balance after the purchases should not go below $0.

2. The number of food items Arnold can buy is limited by the money on the gift card.

Given these factors, we can set up an inequality:

45 - 10x ≥ 0

Now, solve for x:

45 ≥ 10x

Divide both sides by 10:

4.5 ≥ x

Since you can't have a fraction of a food item, the most appropriate domain for the function f(x) is x ≤ 4.

So, the domain of the function is x ≤ 4, which means Arnold can purchase up to 4 food items without exceeding the balance on the gift card.